Summary: We introduce a methodology to quantify the quality of mixing in various systems, including polymeric ones, by adapting the Shannon information entropy. For illustrative purposes we use particle advection of two species in a two‐dimensional cavity flow. We compute the entropy by using the probability of finding a suitable chosen group/complex of particles of a given species, at a given location. By choosing the size of the group to be in direct proportion to the overall concentration of the components in the mixture we ensure that the entropic measure is maximized for the case of perfect mixing, that is, when at each location the component concentration is equal to the corresponding overall component concentrations. The scale of observation role in evaluating mixing is analyzed using the entropic methodology. We also illustrate the effect of initial conditions on mixing in a laminar system, typical in operations involving polymers. magnified image

Abstract-The energy demand imposed by physical exercise on the components of the oxygen transport and utilization system requires a close link between cellular and external respiration in order to maintain ATP homeostasis. Invasive and non-invasive experimental approaches have been used to elucidate mechanisms regulating the balance between oxygen supply and consumption during exercise. Such approaches suggest that the mechanism controlling the various subsystems coupling internal to external respiration are part of a highly redundant and hierarchical multi-scale system. In this work, we present a ''systems biology'' framework that integrates experimental and theoretical approaches able to provide simultaneously reliable information on the oxygen transport and utilization processes occurring at the various steps in the pathway of oxygen from air to mitochondria, particularly at the onset of exercise. This multi-disciplinary framework provides insights into the relationship between cellular oxygen consumption derived from measurements of muscle oxygenation during exercise and pulmonary oxygen uptake by indirect calorimetry. With a validated model, muscle oxygen dynamic responses is simulated and quantitatively related to cellular metabolism under a variety of conditions.

A methodology for rigorous mixing assessment in microchannels is presented. The analysis is based on numerical simulations of flow in different geometries coupled with mixing assessment using entropic measures. The results show enhanced mixing efficiency for the staggered herringbone micromixer by comparison with a mixer with straight diagonal ridges and a lack of mixing in a non-patterned channel. These results are in agreement with published experimental data.

We present a procedure for inducing chaotic mixing based on a non-periodic patterning of the walls making use of the Weierstrass fractal function to generate the locations for the grooves. We show the numerical analysis of flow in three different geometries generated with the Weierstrass function and compare the results with a fourth geometry, quite similar to the staggered herringbone mixer (SHM) of Stroock et al (2002 Science 295 647), for which the patterning is periodic. We evaluate the Lyapunov exponents for massless and non-interacting particles advected by the flow and traced along the channels. We also compute the entropy of mixing for binary mixtures. Finally, we compute generalized (fractal) dimensions associated with the interface of the two fluids. The results show consistently substantial enhancement in mixing efficiency for two of the Weierstrass channels compared to the SHM.

A model for solid agglomerate dispersion in single screw extruders is proposed. The model combines numerical simulations of flow patterns in the metering section of a single screw extruder with a Monte Carlo method of clusters rupture and erosion mediated by a local fragmentation number. Particle size distributions and Shannon entropy are used for mixing characterization. The model is quite general and can be adapted for different polymer-additive systems as well as for different processing equipment.

Abstract.Mixing is an important component in most processing operations including but not limited to polymer processing. Generically, mixing refers to the system capability to reduce composition nonuniformity. Since the entropy is the rigorous measure of disorder or system homogeneity, we will explore in this presentation various ways to employ the entropy to characterize the state of mixing in a multi-component system. The various species can be initially present in the system or they can evolve as a result of a dispersive mixing operation involving a cohesive minor component.

A model of agglomerate break-up, incorporating both rupture and erosion, is employed to predict the dynamics of filler size distribution in a plasticating single screw extruder. Filler spatial distribution along the extruder length was also ascertained and direct comparison of experimental and computational data proved to be satisfactory. The method was also used to investigate the effect of material properties, operating conditions and extruder geometry on the dynamics of agglomerate dispersion along a single screw extruder. Generally, dispersion levels were primarily governed by the magnitude of the hydrodynamic stresses developed in the extruder and the residence time in the melt.

Rigorous measures of mixing are needed to optimise processes and equipment design. In this work, the Shannon entropy, a rigorous measure of mixing universally employed across sciences, is used to quantify laminar mixing in an extruder. The classical unwound channel model of a single screw extruder is used in the present study and the influence of extruder geometry on mixing quality is the focus of this work. Three geometries are analysed: a simple single screw extruder, a single screw extruder with ridges placed diagonally on the bottom of the channel and a single screw extruder with pins on the bottom of the channel. The pin geometry is found to be more efficient at mixing than the other two geometries. The influence of scale of observation on the mixing measure is also studied. As the scale of observation decreases, the entropy decreases.